Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
|
   
Available in electronic format
Available in print format 		Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(e) ISSN 0002-9939(p)

     

Howe duality and the quantum general linear group

Author(s): R. B. Zhang
Journal: Proc. Amer. Math. Soc. 131 (2003), 2681-2692.
MSC (2000): Primary 17B37, 20G42, 17B10
Posted: December 30, 2002
MathSciNet review: 1974323
Retrieve article in: PDF
This article is available free of charge

Abstract | References | Similar articles | Additional information

Abstract: A Howe duality is established for a pair of quantized enveloping algebras of general linear algebras. It is also shown that this quantum Howe duality implies Jimbo's duality between ${\mathrm U}_q({\mathfrak{gl}}_n)$ and the Hecke algebra.

References:

[APW]
Andersen, H. H.; Polo, P.; Wen, K. X., Representations of quantum algebras. Invent. Math. 104 (1991) 1-59. MR 92e:17011

[CP]
Chari, V.; Pressley, A., A guide to quantum groups. Cambridge University Press, Cambridge, 1994. MR 95j:17010

[GZ]
Gover, A. R.; Zhang, R. B., Geometry of quantum homogeneous vector bundles and representation theory of quantum groups. I. Rev. Math. Phys. 11 (1999) 533-552. MR 2000j:81108

[Ho]
Howe, R., Perspectives on invariant theory. The Schur Lectures (1992). Eds. I. Piatetski-Shapiro and S. Gelbart, Bar-Ilan University, 1995. MR 96e:13006

[Ji]
Jimbo, M., Quantum $R$ matrix related to the generalized Toda system: an algebraic approach. In Field theory, quantum gravity and strings (Meudon/Paris, 1984/1985), 335-361, Lecture Notes in Phys., 246, Springer, Berlin (1986). MR 87j:17013

[Ko]
Kostant, B., On Macdonald's $\eta$-function formula, the Laplacian and generalized exponents, Adv. Math. 20 (1976) 257-285. MR 58:5484

[KR]
Kirillov, A. N., Reshetikhin, N., $q$-Weyl group and a multiplicative formula for universal $R$-matrices. Comm. Math. Phys. 134 (1990) 421-431. MR 92c:17023

[Ma]
Mathas, A., Iwahori-Hecke algebras and Schur algebras of the symmetric group. Providence, R.I. : American Mathematical Society (1999). MR 2001g:20006

[Mon]
Montgomery, S., Hopf algebras and their actions on rings. CBMS Regional Conference Series in Math., 82. American Mathematical Society, Providence, RI, 1993. MR 94i:16019

[PW]
Parshall, B.; Wang, J. P., Quantum linear groups. Mem. Amer. Math. Soc. 89 (1991), no. 439. MR 91g:16028

[FRT]
Reshetikhin, N. Yu.; Takhtadzhyan, L. A.; Faddeev, L. D., Quantization of Lie groups and Lie algebras. Algebra i Analiz 1 (1989) 178-206. (Russian) MR 90j:17039

[Ta]
Takeuchi, M., Some topics on ${\rm GL}_q(n)$. J. Algebra 147 (1992) 379-410. MR 93b:17055

Similar Articles:

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 17B37, 20G42, 17B10

Retrieve articles in all Journals with MSC (2000): 17B37, 20G42, 17B10

Additional Information:

R. B. Zhang
Affiliation: School of Mathematics and Statistics, University of Sydney, Sydney, New South Wales 2006, Australia
Email: rzhang@maths.usyd.edu.au

DOI: 10.1090/S0002-9939-02-06892-2
PII: S 0002-9939(02)06892-2
Received by editor(s): June 24, 2001
Received by editor(s) in revised form: April 7, 2002
Posted: December 30, 2002
Communicated by: Dan M. Barbasch
Copyright of article: Copyright 2002, American Mathematical Society




AMS and Social Media LinkedIn Facebook Podcasts Twitter YouTube RSS Feeds Blogs Wikipedia